What Does ∫ (dy)^2 Mean in Calculus?

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Discussion Overview

The discussion centers around the meaning and implications of the integral ∫ (dy)², particularly in the context of calculus and stochastic processes. Participants explore its definition, validity, and mathematical interpretation, raising questions about its existence and the nature of squaring differentials.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion over the meaning of ∫ (dy)², noting that it is not commonly found in elementary calculus textbooks.
  • Another participant suggests that dy is independent of y and can be treated as a constant during integration, leading to the conclusion that ∫ (dy)² equals zero.
  • A participant questions the validity of squaring an infinitesimal quantity, seeking clarification on how this operation is defined within the context of integration.
  • One contribution attempts to define ∫ (dy)² as the limit of a sum, proposing that it approaches zero as n approaches infinity, though acknowledging a lack of rigorous mathematical definition.
  • Another participant introduces the idea that the integral may imply a double integral over ℝ², suggesting that the notation could represent a measure issue related to integrating over a line.
  • A later reply references Ito's Lemma, hinting at connections to stochastic calculus without elaborating on its relevance to the integral in question.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the meaning or validity of ∫ (dy)². Multiple competing views and interpretations are presented, with ongoing questions about its mathematical foundation and implications.

Contextual Notes

There is a lack of formal definitions for the integral of (dy)² in standard mathematics, leading to various interpretations and assumptions about its meaning and application. The discussion reflects uncertainty regarding the treatment of differentials and their properties in integration.

kingwinner
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What is ∫ (dy)^2 ?

Fact:
Let a<b.
b
∫ (dy)2 = 0
a

=====================

1) First of all, I'm having trouble understanding the meaning of the above integral. I tried searching my elementary calculus textbooks, but I really couldn't find ANY integral like this. I have only seen integrals like ∫ (dy) in my calculus books and so I have no idea how they can have (dy)2 which makes no sense to me...

2) Why is the above fact true? How can we rigorously prove it?


Hopefully someone can clarify this.
Thank you!
 
Last edited:
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I guess a,b are real numbers. Where did you see this integral?
 


Yes, a and b are real numbers, a<b.

I saw this integral in my first study of stochastic calculus and brownian motions.
It says...from ordinary integration, we have
b
∫ (dy)2 = 0
a

By ordinary, I'm assuming it's referring to standard elementary calculus, but I just can't recall anything similar to integrals like this in my calculus courses. Does such a thing exist in math? What is the precise definition of the above integral?

Thanks!
 


One work around on this is to consider the fact that dy is independent of y thus it is like a constant when integrating with respect to y. So it is \int (dy)^2=\int cdy

Integrating cdy from a to b will give c(b-a)=dy(b-a). But dy is an infinitesimally small quantity therefore dy(b-a)=0.
 


dy is an infinitesimally small quantity
But then what does squaring an infinitesimally small quantity (i.e. (dy)2) mean? Squaring a differential just doesn't make much sense to me...

Also, I believe integral is defined as the limit of some kind of sum.
Is there an analogous sum being used to define the integral
b
∫ (dy)2 ?
a
 


dy is something like (b-a)/n and the integral is defined as the limit of a sum.
\int_{a}^{b}(dy)^2=lim_{n→∞}\sum _{i=1}^{n}(\frac{b-a}{n})^2

You can see that this limit is zero as n goes to infinity. But all i did here is not rigorously mathematically defined, i mean in math textbooks there is no official definition for the integral of (dy)^2 but an attempt to define such a thing in the most straightforward way is via the limit of sum i gave above.
 
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I don't mean to be pedantic about this, by bringing-in the big machinery, but I can't
come up at this moment with something simpler; tho I will try: using
Green's theorem, one integrates 1-forms over surfaces, so there seems to be
a dimension problem. Maybe it is understood/implied that when using ∫ and (dy)2, that we are actually using a double integral, and using dy repeatedly as the dummy variable
so that ∫(dy)2 can maybe be understood as the integral over ℝ2
of the function f==1 along the x-axis , i.e., a line has measure zero as a subset of ℝ2 . Or, more simply, a line has area=0 .

Maybe someone else can double-check.
 

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